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We develop two types of integral formulas for the perimeter of a convex body K in planar geometries. We derive Cauchy-type formulas for perimeter in planar Hilbert geometries. Specializing to H 2 we get a formula that appears to be new. In the projective model of H 2 we have P = 1 R w d'. Here w is the Euclidean length of the projection of K from the ideal boundary point R = (cos',sin') onto the diametric line perpendicular to the radial line to R (the image of K may contain points outside the model). We show that the standard Cauchy formula P = R sinhr d! in H 2 follows, where ! is a central angle perpendicular to a support line and r is the distance to the support line. The Minkowski formula P = Rg 2 din E 2 generalizes to P = 1 42

Integral-Geometric Formulas for Perimeter in $\S^2$, $\H^2$ and Hilbert Planes